Rational numbers are a "ratio" of one value to another. It's common to think of a fraction as a statement of some number of parts of a particular whole. When working with fractions, it's helpful to think about how to define that "whole" so that various fractional parts can be seen on a common scale. Note 4

To help you visualize this, in this section you will learn how to represent fractions with Cuisenaire Rods and then see how to use the rods to perform operations with fractions.

Here is a set of Cuisenaire Rods:

In order to represent fractions with these rods, you need to choose one rod to serve as a unit (in other words, to represent the whole, or value "1"). The rule to follow is that you must also be able to represent the rod you choose with at least one single-color "train" of the same length, built out of shorter rods. This way you will be able to use the rods to do computations with fractions.

For example, if you want to do computations with halves, the shortest rod you can use to represent "1" is red. That's because you can make a two-car, one-color train out of white rods that is the same length as a red. In this case, each white represents a half:

The next-longest rod that has a two-car, one-color train is the purple rod, and that rod has an all-red train, as well as an all-white train:

The next-longest rod to satisfy the requirement is the dark-green rod, and it also has an all-red train, as well as a light-green and a white train. Notice that the halves in this case are light-green rods. If we name the dark-green rod 1, then the light-green rod is 1/2, the red rod is 1/3, and the white rod is 1/6.

In fact, we could show that every rod that has a two-car, one-color train also has an all-red train. This means that in order to represent halves using rods, the rod length must be divisible by 2, which in our original Cuisenaire configuration is the red rod.